Error-Correcting Codes for Labeled DNA Sequences
Dganit Hanania, Eitan Yaakobi
TL;DR
The paper tackles recovering the original labeling sequence on DNA molecules from noisy readings that may include deletions, insertions, or substitutions. It extends the labeling capacity framework to error-prone channels by developing two regimes: (i) using the full set of length-$2$ labels and (ii) using a minimal length-$2$ label set that still achieves maximal capacity ($\\phi(q)$ labels). It provides both concrete bounds and explicit encoders: a derivative-based construction with lower bounds and sphere-packing-type upper bounds for the all-labels case, a Tenengolts-based systematic encoder for the minimal-label-set case, and a substitution-correcting scheme via Hamming-code cosets with a second explicit encoder. Collectively, these results enable efficient, linear-time encoding and decoding for robust labeling of DNA sequences under single deletion/insertion and single substitution errors, with applicability to general alphabets beyond DNA.
Abstract
Labeling of DNA molecules is a fundamental technique for DNA visualization and analysis. This process was mathematically modeled in [1], where the received sequence indicates the positions of the used labels. In this work, we develop error correcting codes for labeled DNA sequences, establishing bounds and constructing explicit systematic encoders for single substitution, insertion, and deletion errors. We focus on two cases: (1) using the complete set of length-two labels and (2) using the minimal set of length-two labels that ensures the recovery of DNA sequences from their labeling for 'almost' all DNA sequences.